As you said correctly, the weight is defined by mass times gravity. Since you can change the gravity and all the forces independently of the object settings, defining a weight would mess with that. The definition has to be the mass, so forces can calculate their influence on the object. In a way the mass with gravity present is the same as weight, just scaled. Since mass is mostly important relative to each other, you can interpret it as weight if you so must. Forces need to be adjusted accordingly though.
What you are trying to achieve there looks really hard to simulate though. Good luck 🙂 let us know how it goes.
Little tip: the simulations are approximations and some things are far off from reality. Rigid and soft body interactions depend also on the mass of the softbody. In real life I'd say the mass of the surface is barely important and only the elasticity. The relative mass of softbody and rigid body is in the simulation very important to how well the rigid body is held up.